A capillary tube of radius 1 mm is kept vertical with the lower end in water. (a) Find the height of water raised in the capillary. (b) If the length of the capillary tube is half the answer of part (a), find the angle `theta` made by the water surface in the capillary with the wall.

Explanation

To solve the problem step by step, we will break it down into two parts as given in the question.

### Part (a): Finding the height of water raised in the capillary tube

1. **Identify the formula for capillary rise**:
The height of liquid raised in a capillary tube can be calculated using the formula:
\[ h = \frac{2s \cos \theta}{\rho r g} \]
where:
- \( h \) = height of the liquid column
- \( s \) = surface tension of the liquid
- \( \theta \) = angle of contact
- \( \rho \) = density of the liquid
- \( r \) = radius of the capillary tube
- \( g \) = acceleration due to gravity

2. **Substitute the known values**:
Given:
- Radius \( r = 1 \text{ mm} = 1 \times 10^{-3} \text{ m} \)
- Surface tension \( s = 0.075 \text{ N/m} \)
- Angle of contact \( \theta = 0^\circ \) (for water in glass)
- Density of water \( \rho = 1000 \text{ kg/m}^3 \)
- Acceleration due to gravity \( g = 10 \text{ m/s}^2 \)

Now substituting these values into the formula:
\[ h = \frac{2 \times 0.075 \times \cos(0)}{1000 \times (1 \times 10^{-3}) \times 10} \]

3. **Calculate \( h \)**:
Since \( \cos(0) = 1 \):
\[ h = \frac{2 \times 0.075 \times 1}{1000 \times 1 \times 10^{-3} \times 10} \]
\[ h = \frac{0.15}{10} = 0.015 \text{ m} = 1.5 \text{ cm} \]

### Part (b): Finding the angle \( \theta \) made by the water surface with the wall

1. **Determine the new height of water in the capillary**:
The length of the capillary tube is half the height calculated in part (a):
\[ h' = \frac{h}{2} = \frac{1.5 \text{ cm}}{2} = 0.75 \text{ cm} = 0.0075 \text{ m} \]

2. **Use the capillary rise formula again**:
We will use the capillary rise formula again to find the angle \( \theta \):
\[ h' = \frac{2s \cos \theta}{\rho r g} \]
Substituting \( h' = 0.0075 \text{ m} \):
\[ 0.0075 = \frac{2 \times 0.075 \cos \theta}{1000 \times (1 \times 10^{-3}) \times 10} \]

3. **Rearranging to find \( \cos \theta \)**:
\[ 0.0075 = \frac{0.15 \cos \theta}{10} \]
\[ 0.0075 \times 10 = 0.15 \cos \theta \]
\[ 0.075 = 0.15 \cos \theta \]
\[ \cos \theta = \frac{0.075}{0.15} = 0.5 \]

4. **Calculate \( \theta \)**:
\[ \theta = \cos^{-1}(0.5) = 60^\circ \]

### Final Answers:
- (a) The height of water raised in the capillary is **1.5 cm**.
- (b) The angle \( \theta \) made by the water surface with the wall is **60 degrees**.